Over this disagreement looms the monthlong August recess, which McConnell has declared is the unofficial deadline for getting the BCRA passed. After that, McConnell warned, Republicans would have to work with Democrats on a short-term stabilization bill for the individual insurance market.

Because of the stalled out negotiations, 10 GOP senators sent a letter asking McConnell to delay the start of the August recess so Republicans could work on their agenda, including the BCRA.

The Trump administration also favors delaying the recess until a bill is done, as evidenced by Trump's tweet and by the White House chief of staff, Reince Priebus, who told "Fox News Sunday" that a bill needed to get done.

"Whether it'd be before August recess or during August recess, the president expects the Senate to fulfill the promises it made to the American people," Priebus said.